When you are using the route map (https://www.jetblue.com/route-map), make sure you select the Non-stop and Mint filters.

Otherwise, it lists all the possible Mint routes you could take non-stop. For instance, DEN->AUA is listed as a Mint Route when you don't have the Non-stop filter on, because you could fly DEN-JFK-AUA, and JFK-AUA has weekly Mint flights.

As for BOS-LIR and JFK-LIR (and many Caribbean routes for that matter), only one flight weekly is Mint (Saturdays) and it is often sold out. BOS-LIR and JFK-LIR on Jan 29 have Mint seats available. Click on the actual date when you use the Best Fare Finder. The Best Fare Finder is just that - a "Best Fare" finder. Naturally, the best fare is never the Mint Fare, as Core will always be cheaper.